A030011 Inverse Euler transform of {1, primes}.

1, 1, 1, 1, 1, 1, -1, -2, -3, -2, 4, -1, 5, 3, -4, -5, -9, 3, -3, 15, 19, 0, 6, -39, -27, -22, 5, 57, 50, 107, -49, -96, -142, -213, 83, 138, 468, 365, 0, -327, -1215, -618, -526, 957, 2572, 1831, 1673, -2820, -4516, -6155, -3880, 5998, 9282, 18213, 7414

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

Examples

			(1-x)^(-1) * (1-x^2)^(-1) * (1-x^3)^(-1) * (1-x^4)^(-1) * (1-x^5)^(-1) * (1-x^6)^(-1) * (1-x^7) * (1-x^8)^2 * ... = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + ... .
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000040, A030010.

Programs

Mathematica

pp = Prepend[Prime[Range[n = 100]], 1]; s = {};
For[i = 1, i <= n + 1, i++, AppendTo[s, i*pp[[i]] - Sum[s[[d]]*pp[[i - d]], {d, i - 1}]]];
Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i,
n + 1}] (* Jean-François Alcover, May 10 2019 *)

Formula

Product_{k>=1} 1/(1-x^k)^{a(k)} = 1 + x + Sum_{n>=1} prime(n) * x^(n + 1).